Both are Exchange 2010 one is SBS 2011 and the other is Exchange Enterprise Product.

e000035e - The Backup Exec Agent for Microsoft Exchange was not used to create the last full backup of this database. You must use the Exchange Agent to run a full backup before you run a differential or incremental backup.

13.0 Rev 5204 (64-bit)
Backup Full works fine it is the incremental backups that fail.

I cannot find how to install the Agent for Exchange I found the Remote Agent for Windows but not an Exchange Agent.

Also, if there are other backups running during this time or between your fulls and incrementals, etc. then this will cause them to fail as well. If you happen to be using an online backup or other backup solution in addition to BE, please disable those. Then ensure that Exchange maintenance is not running during the backup windows.

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Event ID 9782
Exchange VSS Writer (instance b91bec6c-81fe-427f-9f13-cb97cd625812:134) has completed the backup of database 'Public Folder Database 2008466347' with errors. The backup did not complete successfully, and no log files were truncated for this database.

Exchange VSS Writer (instance 9c7d59b7-951a-4b3d-afec-895c6f92c760:133) has completed the backup of database 'Mailbox Database 1600752793' with errors. The backup did not complete successfully, and no log files were truncated for this database.

I would look into the following:
All Windows updates installed
All Exchange updates installed
All BE updates installed on the Media Server and the Remote Agent
Using Symantec Tape Library drivers
AV is turned off
Separate Exchange DB backups from system backups

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